Recitation 19

Example 1: Determine whether the series is convergent or divergent. If it is convergent, find its sum. (a) \sum\frac{1+3^n}{2^n}; (b) \sum\frac{1}{1+(2/3)^n}; (c) \sum (3/5^n+2/n).

Hint: (a) (b) Find the limit of a_n. (c) If \sum a_n is convergent and \sum b_n is divergent, then \sum (a_n + b_n) is divergent.

Example 2: Determine whether the series \sum_{n=1}^{\infty}\frac{3}{n(n+3)} is convergent or divergent by expressing s_n as a telescoping sum. If it is convergent, find its sum.

Hint: Note that \frac{3}{k(k+3)} = \frac{1}{k} - \frac{1}{k+3}. We have s_n = \sum_{k=1}^n\frac{3}{k(k+3)} = \sum_{k=1}^n(\frac{1}{k} - \frac{1}{k+3}) = 1/1 + 1/2 + 1/3 - 1/(n+1) - 1/(n+2) - 1/(n+3).

Example 3: Find the values of x for which the series \sum_{n=0}^\infty (-4)^n(x-5)^n converges. find the sum of the series for those values of x.

Hint: This is a geometric series with initial term 1 and common ratio -4(x-5). It is convergent if and only if -1 < -4(x-5) < 1, i.e., 19/4 < x < 21/4. When it converges, the sum is 1 / (1 - (-4)(x-5)) = 1/(4x-19).

Example 4: Use the integral test to determine whether the series \sum_{n=1}^\infty \frac{n}{n^2+1} is convergent or divergent.

Hint: Consider f(x) = \frac{x}{x^2+1} for x > 1. Check it is continuous, positive and ultimately decreasing.

Example 5: Determine whether the series is convergent or divergent. (a) 1 + 1/8 + 1/27 + 1/64 + 1/125 + \ldots; (b) \sum_{n=1}^\infty\frac{1}{n^2+4}; (c) \sum_{n=2}^\infty\frac{1}{n\ln n}.

Hint: Consider f(x) = 1/x^3, f(x) = \frac{1}{x^2+4}, f(x) = \frac{1}{x\ln x} for (a), (b), (c) respectively. Check they are continuous, positive and ultimately decreasing.

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